Unit 5 Function Operations

1 1 Unit 5 Function Operations (Book sections and ) NAME _____ PERIOD _____ Teacher _____ 2 Learning Targets Function Operations 1. I can perform Operations with functions . 2. I can evaluate composite functions Function Composition 3. I can write Function rules for composite functions Inverse functions 4. I can graph and identify domain and range of a Function and its inverse. 5. I can write Function rules for inverses of functions and verify using composite functions .. 3 Function Operations Date: _____ After this lesson and practice, I will be able perform Operations with functions . (LT1) evaluate composite functions . (LT2) Having studied how to perform Operations with one Function , you will next learn how to perform Operations with several functions . Function Operation Notation Addition: (f + g) = f(x) + g(x) Multiplication: (f g) = f(x) g(x) Subtraction: (f - g) = f(x) - g(x) Division fg" # $ % & ' x()=f(x)g(x),g(x) 0 The domain of the results of each of the above Function operation are the _____-values that are in the domains of both _____ and _____ (except for _____, where you must exclude any _____-values that cause _____.)

2 (Remember you cannot divide by zero) Function Operations (LT 1) Example 1: Given =3 +8 and =2 12, find h(x) and k(x) and their domains: a) = + and b) =2 Example 2: Given = ! 1 and = +5, find h(x) and k(x) and their domains: a) = b) =!(!)!(!) 4 Your Turn 1: Given =3 1, =2 ! 3, and =7 , find each of the following functions and their domains. a. + ( ) b. ( ) c. ( ) ( ) d. !! Composite functions (LT 2) Let s explore another Function operation using a familiar topic money! Example 3: A store offers a 20% discount on all items and you also have a $3 coupon. Suppose you want to buy an item that originally costs $30. If both discounts can be applied to your purchase, which discount should you apply first? Does it matter? a) 20% then $3 b) $3 then 20% This example demonstrates the idea of _____ functions . Definition 1: Composition of functions is created when the output of one Function becomes the input of another Function .

3 The composition of Function f with Function g is written as _____or _____ and is read as f of g of x The composition of Function g with Function f is written as _____or _____ and is read as g of f of x When evaluating a composite Function , evaluate the _____ Function first. Example 4: Let =2 ! 5 and = 3 +1. Find a. 2 b. 3 This is read g of f of -3 5 Your Turn 2: Let = ! and = 2 +7. Find: a. 4 b. 2 Example 5: Let s return to the shopping example. Let the price of the item you want to purchase be x dollars. Use composition of functions to write two functions : one Function for applying the 20% discount first, and another Function for applying the $3 coupon first. ($50 item) Percent then coupon Coupon then percent How much more is any item if the clerk applies the $3 coupon first to a $50 purchase? FINAL CHECK: Learning Target 1: I can perform Operations with functions . 1. Let f(x)=5x2 1 and g(x)=9x.

4 Find and simplify each Function below. State the restriction to the domain in part c. Show all work. a. g(x) 2f(x) b. !!f(x) g(x) c. !!g(x)f(x) _____ _____ _____, !!x ____ 6 FINAL CHECK: (Cont) Learning Target 2: I can evaluate composite functions . 2. Let f(x)=2x2+5x 1 and g(x)=4x+2. Find and simplify each Function below. Show all work. a. f(g( 3)) b. g(f( 5)) _____ _____ 3. Let f(x)=15x 3 and g(x)= 5x+8. Find and simplify each Function below. Show all work. a. f(g(2)) b. g(g( 3)) _____ _____ Practice Assignment I can use perform Operations with Function . (LT1) I can evaluate composite functions . (LT2) o Worksheet on the next page (for both LT 1 and LT 2) (Answers Practice ) 7 Practice 7- 6 Function Operations 1.

5 A boutique prices merchandise by adding 80% to its cost. It later decreases by 25% the price of items that don t sell quickly. a. Write a Function (x) to represent the price after the 80% markup. b. Write a Function g(x) to represent the price after the 25% markdown. c. Use a composition Function to find the price of an item after both price adjustments that originally costs the boutique $150. d. Does the order in which the adjustments are applied make a difference? Explain. Let (x) = 4x 1 and g(x) = 2x2 + 3. Perform each Function operation and then find the domain. 2. (x) + g(x) 3. (x) g(x) 4. (x) g(x) 5. ()()fxgx 6. g(x) (x) 7. ()()gxfx Let (x) = 3x + 2, g(x) = 5x, h(x) = 2x2 + 9, and j(x) = 5 x. Find each value or expression. 8. (f j)(3) 9. (j h)( 1) 10. (h g)( 5) 11. (g f)(a) 12. (x) + j(x) 13. (x) h(x) 14. (g f)( 5) 15. (f g)( 2) 16. 3 (x) + 5g(x) 17. g(f (2)) 18. g(f (x)) 19. f(g(1)) Let g(x) = x2 5 and h(x) = 3x + 2.

6 Perform each Function operation. 20. (h g)(x) 21. g(x) h(x) 22. 2g(x) + h(x) 23. A department store has marked down its merchandise by 25%. It later decreases by $5 the price of items that have not sold. a. Write a Function (x) to represent the price after the 25% markdown. b. Write a Function g(x) to represent the price after the $5 markdown. c. Use a composition Function to find the price of a $50 item after both price adjustments. d. Does the order in which the adjustments are applied make a difference? Explain. 8 More Practice #1 1) Adding and Subtracting functions . Let f(x) = - 2x + 6 and g(x) = 5x 7. a) Find f + g and it s domain b) Find f g and it s domain 2) Let f(x) = 5x2 4x and g(x) = 5x + 1. a) Find f + g and it s domain b) Find f g and it s domain 3) Multiplying and Dividing functions .

7 Let f(x) = x2 + 1 and g(x) = x4 1. a) Find (f g) and it s domain b) Find and it s domain c) Find f(g(2)) d) g(f(- 2) 4) Let f(x) = 6x2 + 7x - 5 and g(x) = 2x 1. a) Find (f g) and it s domain b) Find and it s domain c) Find f(g(2)) d) g(f(- 2)) 6) A store is offering a 10% discount on all items. In addition, employees get a 25% discount. a) Write a composite Function to model taking the 10% discount first. b) Write a composite Function to model taking the 25% discount first. c) If you were an employee, which would you prefer? fg" # $ % & ' x() fg" # $ % & ' x() 9 More Practice: #2 1) Given f(x) = x + 2 and g(x) = 8x- x2, find the following.

8 A) h(x) = f(x) + g(x) _____ b) h(x) = 3f(x)- g(x) _____ c) h(x) = f(x) g(x) _____ d) h(x) = g(x) + f(x) _____ e) h(x) = g(x) f(x) _____ f) 3 _____ g) g(f(5)) _____ 2) Given f(x) = x2 + 2 and g(x) = 3x- 5 a) f(6) + 3g(2) = b) 2g(6) = c) f(- 4) = d) 5 31g= e) g(f(- 2)) = f) . 1 g) g(f( )) = 10 More Practice #3: Practice with Composite functions using the same f(x) = x2 - 4x + 1 and g(x)= 12x + 3.

9 Remember, work from the inside - out! 1) 3 2) f(g(- 1)) = 3) )61g(f = 4) 4(f(2)) = 75) g(- 1)- 2f(- 3)= 6) g(f( )) = More Practice #4: Function Notation 1. Please find the indicated value of f(x): a. f(x) = -x2 - 3x + 2, 3f(-2) b. f(x)=12x2 4,f(1) _____ _____ Operations with functions 2. Given: f(x) = x2 + 4 & g(x) = 2x - 1, evaluate: a. f(x) + g(x) = b. g(x) - 2f(x) = c. f(g(5)) = a. _____ b. _____ c. _____ d. f(x) * g(x) = e. 3= f. g(f(4)) = d. _____ e. _____ f. _____ 11 Function Composition Date: _____ After this lesson and practice, I will be able write Function rules for composite functions (LT 3). Warm Up: Let =3 ! 4 and =2 +1. Find a. 3 b. 2 In the previous lesson, we found specific values of compositions of two functions .

10 In this lesson, we will find a general formula for Function compositions . In evaluating Function compositions , we substituted the value of x into the input Function , then used the output from that Function as the input on the next Function . With Function composition, the rule of the input Function is used as the input into the next Function . Example 1: Given =3 +1 and =2 +3, find a. b. c. Example 2: Given = !+1 and = +3, find a. b. c. 12 Example 3: Given = !+2 +3 and = 2, find a. b. c. g(g(x)) Example 4: A store offers a 15% discount on all items and you also have a $10 coupon. Both discounts can be applied to your purchase. Write a Function rule, for using the coupon and a Function rule, for using the 15% discount. Write a Function composition rule for using the $10 coupon first and then taking the 15% discount and another rule for using the 15% discount first and then taking the $10 coupon.